Projective plane

Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic.

The term "incidence" is used to emphasize the symmetric nature of the relationship between points and lines.

Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other.

Let K3 denote the set of all triples x = (x0, x1, x2) of elements of K (a Cartesian product viewed as a vector space).

For any nonzero x in K3, the minimal subspace of K3 containing x (which may be visualized as all the vectors in a line through the origin) is the subset of K3.

The minimal subspace of K3 containing x and y (which may be visualized as all the vectors in a plane through the origin) is the subset of K3.

The projective plane over K, denoted PG(2, K) or KP2, has a set of points consisting of all the 1-dimensional subspaces in K3.

A subset L of the points of PG(2, K) is a line in PG(2, K) if there exists a 2-dimensional subspace of K3 whose set of 1-dimensional subspaces is exactly L. Verifying that this construction produces a projective plane is usually left as a linear algebra exercise.

[6] By Wedderburn's Theorem, a finite division ring must be commutative and so be a field.

Taking K to be the finite field of q = pn elements with prime p produces a projective plane of q2 + q + 1 points.

If such subplanes existed there would be projective planes of composite (non-prime power) order.

The seventh line consists of all the diagonal points (usually drawn as a circle or semicircle).

In finite desarguesian planes, PG(2, q), Fano subplanes exist if and only if q is even (that is, a power of 2).

An open question, apparently due to Hanna Neumann though not published by her, is: Does every non-desarguesian plane contain a Fano subplane?

The expression "does not meet" in this condition is shorthand for "there does not exist a point incident with both lines".

A commonly used method for carrying out the embedding in this case involves expanding the set of affine coordinates and working in a more general "algebra".

Alternative, not necessarily associative, division algebras like the octonions correspond to Moufang planes.

There is no known purely geometric proof of the purely geometric statement that Desargues' theorem implies Pappus' theorem in a finite projective plane (finite Desarguesian planes are Pappian).

To describe a finite projective plane of order N(≥ 2) using non-homogeneous coordinates and a planar ternary ring: On these points, construct the following lines: For example, for N = 2 we can use the symbols {0, 1} associated with the finite field of order 2.

The collection of fixed points and fixed lines of a collineation form a closed configuration, which is a system of points and lines that satisfy the first two but not necessarily the third condition in the definition of a projective plane.

Using homogeneous coordinates they can be represented by invertible 3 × 3 matrices over K which act on the points of PG(2, K) by y = M xT, where x and y are points in K3 (vectors) and M is an invertible 3 × 3 matrix over K.[10] Two matrices represent the same projective transformation if one is a constant multiple of the other.

A projective plane is defined axiomatically as an incidence structure, in terms of a set P of points, a set L of lines, and an incidence relation I that determines which points lie on which lines.

By interchanging the role of "points" and "lines" in we obtain the dual structure where I* is the converse relation of I.

In the special case that the projective plane is of the PG(2, K) type, with K a division ring, a duality is called a reciprocity.

By the fundamental theorem of projective geometry a reciprocity is the composition of an automorphic function of K and a homography.

It can be shown that a projective plane has the same number of lines as it has points (infinite or finite).

[citation needed] The existence of finite projective planes of other orders is an open question.

Conversely, one can prove that all Steiner systems of this form (λ = 2) are projective planes.

These turn out to be "tamer" than the projective planes since the extra degrees of freedom permit Desargues' theorem to be proved geometrically in the higher-dimensional geometry.

In fact, each i-dimensional object in PG(d, K), with i < d, is an (i + 1)-dimensional (algebraic) vector subspace of Kd+1 ("goes through the origin").

Drawings of the finite projective planes of orders 2 (the Fano plane ) and 3, in grid layout, showing a method of creating such drawings for prime orders
These parallel lines appear to intersect in the vanishing point "at infinity". In a projective plane this is actually true.
The Moulton plane . Lines sloping down and to the right are bent where they cross the y -axis.
The Fano plane. Points are shown as dots; lines are shown as lines or circles.
(Non-empty) degenerate projective planes
Graph of the projective plane of order 7, having 57 points, 57 lines, 8 points on each line and 8 lines passing through each point, where each point is denoted by a rounded rectangle and each line by a combination of letter and number. Only lines with letter A and H are drawn. In the Dobble or Spot It! game, two points are removed. In the SVG file, hover over a line to highlight it.